When matter and antimatter on an NX-class starship collided together, it created an extreme amount of energy within a warp core which was then transferred to conduits, leading to the nacelles outside the ship which contained the warp coils, thus creating a subspace displacement field around a ship, allowing it to travel at warp speed. (ENT: "Cold Front")

In 2152, a group of Mazarite criminals chased Enterprise on Mazarite warships at warp 4.9 in order to capture AmbassadorV'Lar before she could testify against them. In order to sustain their warp velocity, the ships released drive plasma to keep their warp coils from overloading. (ENT: "Fallen Hero")

In an alternate timeline, CaptainT'Pol crashed a Xindi-Reptilian warship that was docked with Enterprise into another, causing massive damage to Enterprise's starboard nacelle. Commander Tucker told her that half the warp coils in the nacelle were fused. Tucker told T'Pol he would have to rebuild the coil assembly from scratch, which would take him a minimum of six months. (ENT: "Twilight")

In 2154, after awakening from an induced coma for three days, Commander Tucker complained that it would take him days to realign the warp coils on Enterprise after Phlox slightly damaged the ship while trying to take it to warp inside a nebula in the Delphic Expanse. (ENT: "Doctor's Orders")

Later on, Enterprise's primary warp coil was destroyed in the fierce battle of Azati Prime, making warp speed impossible for the starship. Captain Jonathan Archer, with no other options to restore warp capability to Enterprise, stole a primary warp coil from the Illyrians in order for the ship to intercept the Xindi superweapon in time to save Earth. (ENT: "Damage")

Shortly afterwards, T'Pol suggested to Commander Tucker that she could order him to go to sleep, since he had not slept for two days. Tucker told her he couldn't, since Enterprise was full of holes and the only thing allowing them to travel at warp speed was a stolen warp coil. (ENT: "The Forgotten")

William T. Riker considered the warp coil the most important invention of the past two hundred years. Before the coils were invented, "Humans were confined to a single sector of the galaxy". (TNG: "A Matter of Time")

Later that year, Voyager's warp coils were in need of maintenance after the ship narrowly escaped from the Hirogen and a black hole. (VOY: "Hunters")

In a holographic simulation of Voyager's crew created by Seven of Nine, Neelix suggested that re-phasing the warp coils would increase power to the nacelles. (VOY: "One")

In 2375, the crew of Voyager stole a transwarp coil from a Borg sphere and used it in place of its warp coils to generate a transwarp corridor propelling the ship at much faster speeds than a regular warp coil could. The transwarp coil ended up taking Voyager 20,000 light years closer to the Alpha Quadrant before eventually giving out. (VOY: "Dark Frontier")

In 2377, USS Voyager landed on a planet to make repairs. Among the repairs was the removal and maintenance of several warp coils via shuttle. (VOY: "Nightingale")

Curiously, ENT: "Damage" seemed to indicate that a warp coil was a small, portable device built directly into the warp core much like a Borgtranswarp coil. However, in the episode, it was specifically referred to as a "primary warp coil."